On spatial thinning-replacement processes based on Voronoi cells

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We introduce a new class of spatial-temporal point processes based on Voronoi tessellations. At each step of such a process, a point is chosen at random according to a distribution determined by the associated Voronoi cells. The point is then removed, and a new random point is added to the configuration. The dynamics are simple and intuitive and could be applied to modeling natural phenomena. We prove ergodicity of these processes under wide conditions.